The Stacks project

Lemma 52.15.6. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

  1. $A$ has a dualizing complex and is complete with respect to $f$,

  2. for every prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > 2$.

Then the completion functor

\[ \textit{Coh}(\mathcal{O}_ U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_ U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

is fully faithful on the full subcategory of finite locally free objects.

Proof. Follows from Lemma 52.15.5 and Local Cohomology, Proposition 51.10.1. $\square$

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